The present invention relates to a method of numerically simulating physical phenomena, such as mechanical response, heat, viscosity, friction, and electromagnetic effects, occurring in a physical system of interest. More particularly the present invention relates determining physical phenomena in systems which exhibit discontinuous response to physical stimuli, which are subjected to discontinuous physical stimuli, or both.
Numerical simulation using a computer is widely used in industry as a method of simulating the physical response of a physical system to a physical stimulus. By using numerical simulation, it is often possible to reproduce a physical phenomenon, and determine therefrom design principles and parameters. Such practice reduces the number of actual experiments required to reach a final product design, resulting in considerable savings of time and money as well as the possibility of improved functionality of the desired product.
Another application of numerical simulation of physical systems is to provide input to a system controlling a process or action which optimizes that function by determining the current control forces required to obtain a desired future state within constraints established by the nature of the function. An example occurs in robotics, where the desired function can be smooth and rapid motion. A common approach toward determining the control forces for a desired motion involves position and velocity feedback control, but such systems often exhibit jerky motion, or hunting around the desired motion. Control of such systems can be beneficially guided by simulation of the actual dynamics of the robotic system. Similar requirements appear in many mechanical systems, in chemical process control, and in many electronic devices.
The entire process of numerical simulation using a computer can be described as using a computer to transform physical data (material, structure, speed, charge, temperature, etc.) describing a given physical system into physical data describing the behavior of that physical system as a function of time and/or applied conditions. The principle of numerical simulation is to numerically solve equations which are known to provide a sufficiently accurate description of the physical response of a given physical system using a computer. Such equations are usually ordinary or partial differential equations, but integral equations, integrodifferential equations, and many other expressions known to those skilled in the art can appear.
The physical response of a physical system to a physically realizable influence must produce solutions which obey physical continuity and conservation laws. In treating such problems using numerical simulation, time, space, and indeed all system parameters are treated by replacing the true continuous functions with a discretized approximation. In reality all functions in numerical simulation using a computer are discretized owing to the finite word length the computer uses in calculation. Adopting a discrete representation of time and space, thereby restricting physical position to a finite number of `nodes` and time to an integral number of `time steps`, usually allows the equations describing the physical response of a physical system to be simplified to a set of linear algebraic equations. (Not all numerical simulation is directed toward time-dependent behavior. Such matters as distribution of stress under a given set of external constraints can also be treated, usually as boundary-value or mixed problems.) Many approaches to solution of such systems exist, finite-element and finite difference methods among them.
The aforementioned numerical simulation techniques, however, often fail to provide a physically accurate simulation of the physical response of a discontinuous physical system. Such a system can exhibit a discontinuous physical response to a continuous physical stimulus, or can be subjected to a discontinuous physical stimulus. An example of such effects is provided by motion of a body free, save for friction, along the surface of a plate. The discontinuity in this problem is caused by the fact that stiction, or the static coefficient of friction, is larger than is the dynamic coefficient of friction (commonly simply called friction).
Assume the above system starts at rest. If the plate is subjected to an acceleration small enough that the stiction of the body to the plate is greater than the force applied to the body through its contact with the plate, the body will not move relative to the plate. If the force on the body is just large enough that stiction is overcome, there will be relative motion, but that relative motion will be driven by the difference between the force applied to the body through its contact with the plate minus the dynamic friction associated with moving the body across the plate. As stiction is greater than friction, the body experiences a discontinuous change from no net force relative to the plate to a sizable force relative to the plate.
Such discontinuous driving forces present a problem which prevents most such problems from being properly treated using conventional numerical simulation techniques. Simply put, there is no well-founded method within classical analysis to treat such a discontinuity within a discretized representation of the physical system and consistently arrive at a physically correct solution for the system response. Discontinuous differential equations are usually intrinsically nonlinear, and the application of standard analysis to such equations leads to complicated and detailed limit processes which are difficult to treat, either analytically or numerically. As a result, conventional numerical simulation often leads to physically erroneous results. For example, in the frictional example described above, despite the discontinuity in the relative driving force the dependence of the position of the block relative to the plate is still a continuous function of time--it is not possible for the body to suddenly move from one position to a different position without traversing the distance between. However, this clear physical constraint is commonly violated in this class of systems when conventional numerical simulation techniques are applied.
Many real-world examples of such discontinuous physical systems exist. In addition to friction and various analogs thereto, systems involving impact, shock, phase change, combustion fronts, detonation waves, and many more commonly include some component of discontinuity. Conventional numerical simulation of such problems generally yields ambiguous method-dependent results. This limitation is usually addressed through ad hoc introduction of empirical information from experiment. Such ad hoc adaptations force a certain class of behavior on the physical system being simulated, even when the actual behavior of the physical system enters a qualitatively different regime. As a result, application of conventional numerical simulation techniques to such problems is often misleading.
Owing to the common occurrence and importance of discontinuous physical systems, and the significant difficulties which appear in trying to simulate their physical behavior using conventional numerical simulation techniques, there is a clear need to develop new techniques which address such problems and allow their simulation with a reasonable degree of confidence in the resulting solution. Potential applications include design and evaluation of physical systems under conditions experienced in their intended function, and as part of a control system to control processes having fundamental discontinuities.